## Elliptic Functions

### Early Days

Elliptic functions are now understood in several different ways. The most classical way is to consider the inverse of an elliptic integral. Elliptic integrals were studied in detail by Euler and Legendre, but they did not considered the inverse functions. It is Gauss who first noticed the importance of the inverse functions.

 C.F. Gauss unpublished work on elliptic functions started in 1979 and continued through the first decade of the 19th century Gauss studied the inverse function of the arclength of a lemniscate. The arclength integral is a typical elliptic integral. Gauss considered the inverse function of this integral as an analogue of the trigonometric (sine and cosine) functions, discovered a number of essential properties of these functions, and even arrived at the noion of elliptic theta functions and modular functions. Some of these results were later rediscovered by Abel and Jacobi independently.

The unpublished work of Gauss was rediscovered (and largely extended) by Niels H. Abel more than twenty years later. Of course Abel did not know of Gauss' work on elliptic functions. He was led to elliptic functions through his preceding research on algebraic equations, in particular, the ``lemniscate equations'' that Gauss considered in his book on number theory (but without any explicit comment on the relation to elliptic functions).

 N.H. Abel ``Memoire sur une propriété générale d'une classe tres étendue de fonctions transcendents'', submitted to the Academy of Science during his first visit to Paris in 1826, but ``lost'' for many years and finally published in 1841. This paper presents a result that is now known as ``Abel's theorem''. This is a generalization of Euler's results on elliptic integrals, and became a prototype of the subsequent studies on algebraic curves and Riemann surfaces. N.H. Abel ``Recherches sur les fonctions elliptiques'', published in Crelle's journal (J. Reine Angew. Math.) vol. 2 (1827) and vol.3 (1828) In these papers, Abel arrived at the same position as Gauss did, namely the inverse functions of elliptic integrals and the double periodicity. Even some of the notations were the same. That was the true beginning of the studies on elliptic functions in the 19th century.

While Abel was preparing a far extended version of ``Recherchez'', a paper of Jacobi appeared in ``Astronomische Nacharichten''. (This is a journal founded by H.C. Schumacher, who was astronomer and a very intimate frend of Gauss.)

 C.G.J. Jacobi published two papers on elliptic integrals in ``Astronomische Nacharichten'', vol. 123 (1827) and vol. 127 (1827) In this paper, Jacobi reported a theorem on elliptic integrals, first without any proof, then with a proof based on the idea of inversion of elliptic integrals and the double periodicity.

Abel was shocked by this paper of Jacobi. This paper showed that Jacobi was catching the most essential part of elliptic functions. Abel was thus suddenly forced to entere a rivalry with Jacobi. Abel did a counterattack by publishing a more general result in the same journal in 1828, but had to suspend the project for the second part of ``Recherche''. This rivalry, however, was interrupted by Abel's death in 1829. Jacobi continued his researches, studying in particular the theta functions as an independent subject.

 E. Galois mentioned a few results on abelian integrals in the letter written the day before his death in 1832 Galois's last letter mentioned, besides his work on algebraic equations, a few general results on abelian integrals. This is quite surprising, because even Jacobi had just started his researches on hyperelliptic functions.

### Riemann, Weierstrass, and their successors

The double periodicity, already noticed by Gauss, of elliptic functions implies that the most natural stage of the theory of elliptic function is a torus, namely, a Riemann surface of genus one. The notion of Riemann surfaces (or, equivalently, of complex algebraic curves) provides a unified geometric framework for understanding previously known results on elliptic and hyperelliptic integrals. Of course it was Riemann who first proposed this geometric framework.

 G.F.B. Riemann Theorie der Abelischen Funktionen, J. Reine Angew. Math. 54 (1857), 115 - 155

Weierstrass aimed at a rigorous analytical formulation of theory of complex analytic functions. Of particular importance was the justification of infinite series/product expansion. For elliptic functions, Gauss, Abel and Jacobi already made full use of such expansions. Having the infinite product expansion of Euler's gamma function as a prototype, Weierstrass defined his famous sigma function by an infinite product, and reorganized the theory of elliptic functions in his theory of complex analytic functions:

 K. Weierstrass Mathematische Werke, I - VII, Akademsche Verlag 1894 - 1927

Along with these theoretical progress, many mathematicians published monographs and textbooks on elliptic functions in the second half of the 19th century (see F. Klein's ``Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, Springer-Verlag 1926 - 27, Chelsea 1956).